The impact of the solar wind onto the magnetosphere generates an electric field within the inner magnetosphere (r < 10 a; with a the Earth's radius) - the convection field. Its general direction is from dawn to dusk. The co-rotating thermal plasma within the inner magnetosphere drifts orthogonal to that field and to the geomagnetic field Bo. The generation process is not yet completely understood. One possibility is viscous interaction between solar wind and the boundary layer of the magnetosphere (magnetopause). Another process may be magnetic reconnection. Finally, a hydromagnetic dynamo process in the polar regions of the inner magnetosphere may be possible. Direct measurements via satellites have given a fairly good picture of the structure of that field. A number of models of that field exists. A widely used model is the Volland-Stern model
Model Description It is based on two simplifying assumptions: first, a coaxial geomagnetic dipole field B is introduced. Its magnetic field lines can be represented by the shell parameter
with r the distance from the Earth, a the Earth's radius, and θ the co-latitude. For r = a, θ is the co-latitude of the foot point of the line on the ground. L = const is the equation of a magnetic field line, and r = a L is the radial distance of the line at the geomagnetic equator (θ = 90°). Second, it is assumed that the electric field can be derived from an electrostatic potential Φc. Since in a highly conducting electric plasma like the magnetosphere, the electric fields must be orthogonal to the magnetic fields, the electric potential shell is parallel to the magnetic shell. The relation
fulfills that condition. Here L m = 1 sin 2 θ m {\displaystyle L_{m}={\frac {1}{\sin ^{2}\theta _{m}}}} is the separatrix separating the low latitude magnetosphere with closed geomagnetic field lines at θ ≥ θm from the polar magnetosphere with open magnetic fieldlines (having only one footpoint on Earth), and τ the local time. θm ~ 20° is the polar border of the auroral zone. q, Φco, and τco are empirical parameters, to be determined from the observations. Eq.(2) yields for a coordinate system co-rotating with the Earth, its geomagnetic equator being identical with the geographic equator. Since the electric potential is symmetric with respect to the equator, only the northern hemisphere needs to be considered. The general direction of the potential is from dawn to dusk, and Φco is the total potential difference. For a transformation from a rotating magnetospheric coordinate system into a non-rotating system, τ must be replaced by the longitude -λ.
Inner Magnetosphere With the numbers q ~ 2, and Φco and τco increasing with geomagnetic activity (e.g., Φco ~ 17 and 65 kVolt, and τco ~ 0 and 1 h, during geomagnetically quiet and slightly disturbed conditions, respectively), eq.(2) valid at lower latitudes, (θ > θm) and within the inner magnetosphere (r ≤ 10 a) is the Volland-Stern model (see Fig. 1 a)).
The use of an electrostatic field means that this model is valid only for slow temporal variations (of the order of one day or larger). The assumption of a coaxial magnetic dipole field implies that only global scale structures can be simulated. The electric field components are derived from
as
E r = − q r Φ c E θ = 2 q r cot θ Φ c E λ = − 1 r sin θ cot ( λ − λ c o ) Φ c o {\displaystyle {\begin{aligned}&E_{r}=-{\frac {q}{r}}\Phi _{\mathrm {c} }\\&E_{\theta }={\frac {2q}{r}}\cot \theta ~\Phi _{c}\\&E_{\lambda }=-{\frac {1}{r\sin \theta }}\cot(\lambda -\lambda _{\mathrm {co} })\Phi _{\mathrm {co} }\end{aligned}}}
In the presence of the geomagnetic field an electric field is generated in a rotating on frame of reference in order to compensate for the Lorentz force. This is the so-called electric co-rotation field measured by an observer rotating with the Earth. With the simplifying conditions given above its potential is
with Φro = 90 kVolt. The thermal plasma within the inner magnetosphere co-rotates with the Earth. In a non-rotating frame of reference, it reacts to the sum of both fields
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